In the field of portable recoilless weapon systems, accurately measuring dynamic unbalanced parameters is crucial for assessing safety and performance. Traditional testing methods, such as rail-based systems or ballistic pendulums, often fail to capture the complex spatial force systems acting on the device during firing. This limitation can lead to significant errors in evaluating parameters like unbalanced impulse, which directly impact firing accuracy and stability. To address this, we developed a novel six-axis force sensor capable of real-time monitoring of three-dimensional forces and moments during testing. This six-axis force sensor is designed to integrate into existing unbalanced impulse test setups, providing comprehensive data on the force systems encountered. The design focuses on enhancing sensitivity, reducing cross-axis coupling, and ensuring structural robustness under dynamic loading conditions.
The core of our six-axis force sensor is an elastic body structure based on a cross-beam configuration, optimized for high stiffness and minimal interference between measurement channels. The initial design incorporated four fixed blocks, four strain beams, four adapter blocks, eight floating beams, and a loading block, all manufactured from martensitic precipitation-hardening stainless steel (0Cr17Ni4Cu4Nb). This material offers excellent mechanical properties, including an elastic modulus of 196 GPa and a yield strength of 1230 MPa, ensuring durability under high loads. Finite element analysis (FEA) was employed to simulate the structural behavior under various loads, such as forces and moments applied along the X, Y, and Z axes. The strain distribution analysis revealed that stress concentrations occur primarily in the slotted regions of the strain beams, guiding the optimal placement of strain gauges for full-bridge circuits. This initial model served as the foundation for further optimization using response surface methodology (RSM) and multi-objective particle swarm optimization (MOPSO).
The mathematical model of the six-axis force sensor is based on the strain compliance matrix, which relates the applied loads to the output strains. For a six-axis force sensor, the relationship between the input load vector $\mathbf{Q} = [F_x, F_y, F_z, M_x, M_y, M_z]^T$ and the output voltage vector $\mathbf{U}$ can be expressed as $\mathbf{Q} = \mathbf{C}^{-1} (\mathbf{U} – \mathbf{B})$, where $\mathbf{C}$ is the calibration matrix and $\mathbf{B}$ accounts for offsets. In ideal conditions, the output is linear with respect to the input, but practical factors like manufacturing tolerances introduce nonlinearities. The sensitivity of the six-axis force sensor is defined separately for force and moment channels. For a full-bridge strain gauge circuit, the sensitivity $S$ is given by:
$$S = \frac{U_{ES}}{E_0} = \frac{1}{4} G_f (\varepsilon_{1ES} – \varepsilon_{2ES} – \varepsilon_{3ES} + \varepsilon_{4ES}),$$
where $U_{ES}$ is the full-scale output voltage, $E_0$ is the input voltage, $G_f$ is the gauge factor, and $\varepsilon_{iES}$ represents the microstrain at full scale. The overall sensitivity is determined by the lowest sensitivity among the force and moment channels. To quantify the balance between force and moment sensitivities, we define the ratio $S_{F/M} = \left| \frac{S_F}{S_M} – 1 \right|$, where a smaller value indicates better balance. The strain compliance matrix $\mathbf{C}$ captures the linear relationships between inputs and outputs, with diagonal elements representing the main channel sensitivities and off-diagonal elements indicating cross-coupling.
Key performance metrics for the six-axis force sensor include linearity, hysteresis error, and stiffness. Linearity error is the maximum deviation between the calibration curve and the fitted line, expressed as a percentage of full-scale output. Hysteresis error accounts for differences between loading and unloading cycles. Stiffness requirements are specified as $K_{F_x}, K_{F_y} \geq 100\, \text{MN/m}$ for forces and $K_{M_x}, K_{M_y} \geq 100\, \text{kN·m/rad}$ for moments, ensuring minimal deformation under load. The design targets a measurement range of $\pm 500\, \text{N}$ for forces and $\pm 130\, \text{N·m}$ for moments, with overload considerations leading to test conditions of $F_x = F_z = 1500\, \text{N}$ and $M_x = M_z = 200\, \text{N·m}$ during optimization.
To optimize the six-axis force sensor, we conducted a single-factor analysis to identify critical dimensional parameters influencing performance. The parameters included floating beam thickness ($X_1$), strain beam thickness ($X_2$), strain beam height ($X_3$), adapter block thickness ($X_4$), and adapter block height ($X_5$). FEA simulations under varying parameters showed that $X_3$ had the most significant impact, with performance variations up to 57%. Based on this, we defined the factor levels for RSM, as summarized in Table 1.
| Factor | Level 1 | Level 2 | Level 3 |
|---|---|---|---|
| $X_1$ (mm) | 1.40 | 1.55 | 1.70 |
| $X_2$ (mm) | 2.90 | 3.10 | 3.30 |
| $X_3$ (mm) | 18.00 | 20.00 | 22.00 |
| $X_4$ (mm) | 5.60 | 6.85 | 8.10 |
| $X_5$ (mm) | 22.00 | 24.50 | 27.00 |
Using Box-Behnken design, we generated 46 experimental sets and performed FEA to obtain response values $y_1$, $y_2$, $y_3$, and $y_4$, corresponding to strains under $F_x$, $F_z$, $M_x$, and $M_z$ loads, respectively. The response surface models were fitted via least squares regression. For example, the model for $y_1$ (strain under $F_x$) is:
$$y_1(\mathbf{x}) = 2788.6039 – 685.01X_1 – 752.32X_2 – 23.4X_3 + 43.6355X_4 – 42.816X_5 + 44.75X_1X_2 + 12.98X_1X_3 – 2.85X_1X_4 + 1.81X_2X_3 + 6.1X_2X_5 – 1.6X_3X_4 + 0.282X_3X_5 + 57.4167X_1^2 + 71.78X_2^2 – 0.1068X_3^2 – 0.6132X_4^2 + 0.311X_5^2.$$
Similar models were derived for $y_2$, $y_3$, and $y_4$. Statistical analysis confirmed the models’ adequacy, with coefficients of determination ($R^2$) above 0.97 and signal-to-noise ratios exceeding 24, indicating high precision. The ANOVA results showed P-values less than 0.0001, validating the significance of the models for optimization.
We applied multi-objective particle swarm optimization (MOPSO) to the response surface models, with a population size of 40, dimension 5, and 120 iterations. The optimization aimed to maximize sensitivity, minimize $S_{F/M}$, and ensure stiffness constraints. The objective function was weighted as $f(\mathbf{x}) = 0.4f(y_3) + 0.3f(y_4) + 0.2f(y_1) + 0.1f(y_2)$, prioritizing moment channels. The optimized dimensions, compared to the initial design, are shown in Table 2.
| Model | $X_1$ (mm) | $X_2$ (mm) | $X_3$ (mm) | $X_4$ (mm) | $X_5$ (mm) |
|---|---|---|---|---|---|
| Initial | 1.55 | 3.10 | 20.00 | 6.85 | 24.50 |
| Optimized | 1.55 | 2.90 | 18.00 | 7.87 | 24.50 |
The optimization resulted in significant improvements in response values, as detailed in Table 3. The moment channel sensitivities increased by up to 32.24%, and the force channel sensitivities by up to 31.02%, demonstrating the effectiveness of the approach for enhancing the six-axis force sensor performance.
| Item | $y_1$ (με) | $y_2$ (με) | $y_3$ (με) | $y_4$ (με) |
|---|---|---|---|---|
| Initial Model | 199.36 | 74.96 | 513.68 | 641.91 |
| Optimized Model | 223.92 | 98.21 | 679.72 | 778.23 |
| Improvement (%) | 12.31 | 31.02 | 32.24 | 21.24 |
A prototype of the six-axis force sensor was fabricated based on the optimized dimensions and subjected to static calibration using a dedicated setup. The calibration involved applying known loads and measuring the output voltages to derive the calibration matrix. The resulting equation was $\mathbf{Q} = \mathbf{C}^{-1} \mathbf{U}$, with the matrix $\mathbf{C}^{-1}$ given by:
$$\mathbf{C}^{-1} = \begin{bmatrix}
-1.0552 & 0.1607 & -0.5407 & 0.0372 & 0.0989 & 0.5443 \\
-0.2402 & 1.7128 & -0.5867 & 0.1430 & -0.2772 & -0.2214 \\
-0.6191 & -0.7565 & 2.7340 & 0.5191 & -0.1588 & -0.1295 \\
-0.0022 & -0.0112 & 0.0533 & -0.0512 & -0.0090 & -0.0239 \\
0.0242 & -0.0073 & -0.0312 & -0.0232 & -0.0767 & 0.0149 \\
0.0441 & -0.0080 & 0.0245 & -0.0264 & -0.0016 & 0.0320
\end{bmatrix}.$$
The sensitivities and cross-coupling errors after decoupling are summarized in Table 4. The force channel sensitivity was 0.30 mV/V, and the moment channel sensitivity was 1.22 mV/V, with maximum cross-coupling error of 1.01%. This low coupling indicates effective decoupling, essential for accurate measurements in the six-axis force sensor.
| Channel | Sensitivity (mV/V) | Cross-Coupling Error (%) | |||||
|---|---|---|---|---|---|---|---|
| $F_x$ | $F_y$ | $F_z$ | $M_x$ | $M_y$ | $M_z$ | ||
| $F_x$ | 0.61 | — | 0.03 | 0.30 | 0.32 | 0.02 | 0.42 |
| $F_y$ | 0.62 | 0.02 | — | 0.16 | 0.03 | 0.20 | 0.36 |
| $F_z$ | 0.30 | 0.07 | 0.26 | — | 0.11 | 0.13 | 0.31 |
| $M_x$ | 1.47 | 0.52 | 0.06 | 0.94 | — | 0.06 | 0.28 |
| $M_y$ | 1.22 | 0.06 | 0.20 | 0.72 | 0.07 | — | 0.25 |
| $M_z$ | 1.60 | 1.01 | 0.05 | 0.43 | 0.46 | 0.07 | — |
Linearity and hysteresis errors were also evaluated, as shown in Table 5. The maximum linearity error was 1.60% for the $F_z$ channel, and the maximum hysteresis error was 1.03%, also for $F_z$. These values are within acceptable limits for precision measurements, confirming the six-axis force sensor’s reliability.
| Channel | Linearity Error (%) | Hysteresis Error (%) |
|---|---|---|
| $F_x$ | 1.28 | 0.69 |
| $F_y$ | 1.06 | 0.85 |
| $F_z$ | 1.60 | 1.03 |
| $M_x$ | 1.27 | 0.56 |
| $M_y$ | 1.39 | 0.91 |
| $M_z$ | 1.23 | 0.88 |
In conclusion, the optimized six-axis force sensor demonstrates enhanced performance suitable for unbalanced impulse testing. The integration of response surface methodology and multi-objective optimization effectively improved sensitivity and reduced coupling, while the static calibration validated the sensor’s accuracy. This six-axis force sensor provides a robust solution for real-time force system monitoring in dynamic environments, contributing to safer and more reliable weapon system evaluations. Future work could focus on further miniaturization and application in other high-precision fields.